Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,16)(13,18)(14,19)(15,23)(17,21)(22,24);(2,5)(6,8)(9,11,14,12)(10,16,19,20)(13,22,15,21)(17,23,24,18);(9,11,14,12)(10,17,19,24)(13,21,15,22)(16,23,20,18);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,19,20)(13,21,15,22)(17,23,24,18);(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,24,20,17)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #6575439, #6575444, #6575451, … |
| maximal normal subgroup of | #6575439, #6575444, #6575451, … |
| maximal quotient of | #6370991, #6371813, #6371814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53589);